Play had earlier been held up twice due to rain and the threat of thunder and lightning.
Langer, who won the tournament in 2010 at Carnoustie, held a three-shot lead heading into the final day, and looked to have secured the title with another solid round.
American Wiebe, who shared the lead with Langer after the second day, hit a four-under-par 66 on the final round which was good enough for outright second until Langer's meltdown on the 18th.
The pair played two extra holes in near darkness before the decision was made around 2045 GMT to come back on Monday morning to resume the playoff.
America's Corey Pavin, Peter Senior of Australia and South Africa's David Frost were tied for third at six-under par.
(Reporting by Josh Reich; editing by Toby Davis)
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